Scoped negative result: Hurwitz copy floor for fixed-support nonlinear messages

[azide0x37]

Scoped negative result; no new kappa. On the ten Hurwitz E8 lines, fixed-support, deterministic, zero-error messages have minimum charge 10b quaternionic / 40b rational units for b independent bits per port, even with nonlinear encoding and joint decoding. The essential premise is explicit: a q-bit message is charged q times its reader/source pairing rank, additively across messages. This note does not establish that physical pricing premise for arbitrary nonlinear circuits.

The proof extends Douglas R. Colkitt's scalar-total result in #703 by replacing binary rank with conditional transcript injectivity. It retains the predecessor's geometry, charge table and weighted transversals. Arbitrary finite block widths, unequal demands and any finite number of messages are allowed. An explicitly nonlinear code attains the bound. Under the additional inherited Dfc accounting, even zero helper, access, frame, restoration and recursive expenses give D/T <= -40b, so no paid construction or assembly improvement is claimed.

Only research/hurwitz-nonlinear-copy-floor/ is added. Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. The selected construction and upstream/ are unchanged. Seed 42, structural decoder research. The local claim records three candidates, the exact scope and result; no pinned sprint issue exists.

Reproduce with CPython 3.10+ and the standard library:

uv run --offline python -B research/hurwitz-nonlinear-copy-floor/verify.py
uv run --offline python -B research/hurwitz-nonlinear-copy-floor/test_verify.py

Expected main receipt:

PASS_HURWITZ_NONLINEAR_COPY: 1048576 support pairs; 3 attaining codes; 12 rejection controls; conditional floor 10b

Validation passes on CPython 3.10.21, 3.13.15 and optimized 3.14.7. The package checks all 240 roots, 100 singleton pairing ranks, 45 two-line spans, all 1,048,576 support pairs and 33,856 conditional transcripts. Four additional tests compare all 145 geometric obligations with an independent Fraction column algorithm, explicitly invert the nonlinear examples, exhaust all 65,536 one-bit encoders on four ports, and reject receipt/manifest mutations in a relocated isolated optimized replay. A clean committed git archive passes from an unrelated working directory. Selected-record integrity and whitespace checks pass. The unchanged expensive repository-wide and selected cold-replay suites were not repeated; no remote CI or proof-assistant pass is claimed.

The physical width-pricing and stage-accounting assumptions remain unresolved outside the declared model. Adaptive or interactive supports, correlated inputs, errors, cheaper sharing between coordinates and extra nonorthogonal side information are excluded. An explicit boundary example demonstrates why the last restriction matters. Finite replay is not the all-width proof.

Pre-submission overlap checks covered the full live queue through #775, then recent arrivals through #782. They inspected #765's tensor factorization result, #767's Gaussian subclass obstructions and #770's Gaussian weighted-clique ceiling, which have distinct model or geometry scope. Exact predecessor pins and credit are included for #703, Joel Pulikkan / GamingPuzzled's #620, DreamingOfClouds' #446/accounting work and Sagar Thakrar's related #738. No predecessor implementation is imported.

Prepared by Alexander Templeton (@azide0x37) with substantial OpenAI Codex assistance in proof development, implementation, verification and writing. Apache-2.0. No independent human review is claimed.

[Mister-Meeseeks]

100-PR batch review at 83e78085ae9c110a34bff87e71387292d85e47c0.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Hurwitz nonlinear fixed-support message lower bound via conditional transcript injectivity; per-bit pairing-rank pricing is an explicit premise.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.